Linear perturbation theory and structure formation in a Brans-Dicke theory of gravity without dark matter

We investigate the formation of the large-scale cosmic structure in a scalar-tensor theory of gravity belonging to the class of the Brans--Dicke theories. The universe contains baryonic matter alone and neither dark matter nor dark energy. The two arbitrary functions of the scalar field characterizing the kinetic term and the self-interaction potential are set to $W(\varphi)=-1$ and $V(\varphi) = -\Xi \varphi$, respectively, with $\Xi$ a positive constant. In the weak-field limit, the theory reduces to Refracted Gravity, a non-relativistic theory whose modified Poisson equation contains the scalar field $\varphi$ that provides the gravitational boost required to describe the dynamics of galaxies and galaxy clusters without dark matter. In a flat, matter-dominated, homogeneous and isotropic universe the same scalar field $\varphi$ drives the accelerated expansion of the universe and describes the observed redshift evolution of the Hubble-Lema\^itre parameter $H(z)$. However, in the equation of the growth factor of the linear perturbation theory, the form of $V(\varphi)$ makes the coefficient of the source of the gravitational field proportional to $H^{-1}(z)$; therefore the gravitational field is strongly suppressed at early times and structure formation is delayed to redshift $z< 1$, in disagreement with the observation of formed galaxies at much larger redshifts. In addition, the form of $W(\varphi)$ and a linear $V(\varphi)$ imply that $\varphi$ generates twice the gravitational boost on massive particles than on photons, with possible observable consequences on the gravitational lensing phenomenon. It remains to be investigated whether different choices of $W(\varphi)$ and $V(\varphi)$, that can still make the theory reduce to Refracted Gravity in the weak-field limit, might alleviate these problems.